Solving a problem of angiogenesis of degree three

0.27 MB·

by  Anastasios N. Zachos

SOLVING A PROBLEM OF ANGIOGENESIS OF DEGREE
THREE
6
1
ANASTASIOSN.ZACHOS
0
2
Dedicated toProfessors Dr. Alexander O. Ivanov and Dr. Alexey A. Tuzhilin for their

n contributions on minimal trees
a
J
3 Abstract. An absorbing weighted Fermat-Torricelli tree of degree four is a
2 weightedFermat-Torricellitreeofdegreefourwhichisderivedasalimitingtree
structure fromageneralized Gausstreeofdegreethree (weighted fullSteiner
] tree)ofthesameboundaryconvexquadrilateralinR2.Bylettingthefourvari-
C ablepositiveweightswhichcorrespondtothefixedverticesofthequadrilateral
O andsatisfythedynamicplasticityequationsoftheweightedquadrilateral,we
obtain a family of limiting tree structures of generalized Gauss trees which
h. concentrate to the same weighted Fermat-Torricelli tree of degree four (uni-
t versal absorbingFermat-Torricellitree). Thevalues of theresidualabsorbing
a ratesforeachderivedweightedFermat-Torricellitreeofdegreefouroftheuni-
m versalFermat-Torricellitreeformauniversalabsorbingset. Theminimumof
[ theuniversalabsorbingFermat-Torricellisetisresponsibleforthecreationof
a generalized Gauss tree of degree three for a boundary convex quadrilateral

1 derived by a weighted Fermat-Torricelli tree of a boundary triangle (Angio-
v genesisofdegreethree). Eachvaluefromtheuniversalabsorbingsetcontains
0 anevolutionaryprocessofageneralizedGausstreeofdegreethree.
1
3
6
0
. 1. Introduction
1
0
WeshalldescribethestructureofageneralizedGausstreewithdegreethreeand
6
a weightedFermat-Torricellitree of degree four with respect to a boundary convex
1
: quadrilateral A1A2A3A4 in R2.
v
i Definition 1. [5,Section2,pp.2] Atreetopology is a connection matrix specifying
X
which pairs of points from the list A1,A2,A3,A4,A0,A0′ have a connecting linear
r
a segment (edge).
Definition 2. [2, Subsection 1.2,pp. 8][5],[9] The degree of a vertex is the number
of connections of the vertex with linear segments.
LetA ,A ,A ,A befournon-collinearpointsinR2andB beapositivenumber
1 2 3 4 i
(weight) which corresponds to A for i=1,2,3,4.
i
The weighted Fermat-Torricelli problem for four non-collinear points (4wFT
problem) in R2 states that:
1991 Mathematics Subject Classification. 51E12,52A10,52A55,51E10.
Keywordsandphrases. UniversalabsorbingFermat-Torricelliset,UniversalFermat-Torricelli
minimum value, generalized Gauss problem, weighted Fermat-Torricelli problem, weighted
Fermat-Torricelli point, absorbing Fermat-Torricelli tree, absorbing generalized Gauss tree, evo-
lutionarytree.
1
2 ANASTASIOSN.ZACHOS
Problem1(4wFTproblem). Findapoint(weightedFermat-Torricellipoint)A ∈
0
R2, which minimizes
4
f(A )= B kA −A k, (1.1)
0 i 0 i
i=1
X
where k·k denotes the Euclidean distance
By letting B = B = B = B in the 4wFT problem we obtain the following
1 2 3 4
two cases:
(i) If A A A A is a convex quadrilateral, then A is the intersection point of
1 2 3 4 0
the two diagonals A A and A A ,
1 3 2 4
(ii) If A is an interior point of △A A A , then A ≡ A , for i,j,k,l = 1,2,3,4
i j k l 0 i
and i6=j 6=k 6=l.
Thecharacterizationofthe(unique)solutionofthe4wFTprobleminR2 isgiven
by the following result which has been proved in [1] and [10]:
Theorem 1. [1, Theorem 18.37, p. 250],[10]
Let A be a weighted minimum point which minimizes (1.1).
0
(a) Then, the 4wFT point A uniquely exists.
0
(b) If for each point A ∈{A ,A ,A ,A }
i 1 2 3 4
4
k B ~u k>B , (1.2)
j ij i
j=1,i6=j
X
for i,j =1,2,3,4 holds, then
(b ) A does not belong to {A ,A ,A ,A } and
1 0 1 2 3 4
(b )
2
4
B ~u =~0, (1.3)
i 0i
i=1
X
where~u istheunitvectorfromA toA ,fork,l∈{0,1,2,3,4}(WeightedFloating
kl k l
Case).
(c) If there is a point A ∈{A ,A ,A ,A } satisfying
i 1 2 3 4
4
k B ~u k≤B , (1.4)
j ij i
j=1,i6=j
X
then A ≡A . (Weighted Absorbed Case).
0 i
TheinverseweightedFermat-Torricelliproblemforfournon-collinearpoints(In-
verse 4wFT problem) in R2 states that:
Problem 2. Inverse 4wFT problem Given a point A which belongs to the convex
0
hull of A A A A in R2, does there exist a unique set of positive weights B , such
1 2 3 4 i
that
B +B +B +B =c=const,
1 2 3 4
for which A minimizes
0
4
B kkA −A k.
i 0 i
i=1
X
SOLVING A PROBLEM OF ANGIOGENESIS 3
By letting B = 0 in the inverse 4wFT problem we derive the inverse 3wfT
4
problem which has been introduced and solved by S. Gueron and R. Tessler in [6,
Section 4,p. 449].
In 2009, a negative answer with respect to the inverse 4wFT problem is given
in [13, Proposition 4.4,p. 417] by deriving a dependence between the four variable
weightsinR2.In2014,weobtainthe samedependenceofvariableweightsonsome
C2 surfaces in R3 and we call it the ”dynamic plasticity of convex quadrilaterals”
([16, Problem 2, Definition 12, Theorem 1,p.92,p.97-98]).
An important generalization of the Fermat-Torricelli problem is the generalized
Gauss problem (or full weighted Steiner tree problem) for convex quadrilaterals in
R2 which has been studied on the K-plane (Sphere, Hyperbolic plane, Euclidean
plane) in [15].
We mention the following theorem which provide a characterization for the so-
lutions of the (unweighted )Gauss problem in R2.
Theorem2. [1,Theorem(*),pp.328]AnysolutionoftheGaussproblemisaGauss
tree(equally weighted full Steiner tree) with at most two (equally weighted) Fermat-
Torricelli points (or Steiner points) where each Fermat-Torricelli point has degree
three, and the angle between any two edges incident with a Fermat-Torricelli point
is of 120◦.
WeneedtomentionallthenecessarydefinitionsoftheweightedFermat-Torricelli
tree and weighted Gauss tree topologies, in order to derive some important evolu-
tionarystructuresoftheFermat-Torricellitrees(AbsorbingFermat-Torricellitrees)
and Gauss trees (Absorbing Gauss trees) which have been introduced in [17, Defi-
nitions 1-7,p. 1070-1071].
Definition 3. A weighted Fermat-Torricelli tree topology of degree three is a tree
topology with all boundary vertices of a triangle having degree one and one interior
vertex (weighted Fermat-Torricelli point) having degree three.
Definition 4. A weighted Fermat-Torricelli tree topology of degree four is a tree
topology with all boundary vertices of a convex quadrilateral having degree one and
one interior vertex (4wFT point) having degree four.
Definition 5. [5, Subsection 3.7,pp. 6] A weighted Gauss tree topology (or full
Steiner tree topology) of degree three is a tree topology with all boundary vertices of
aconvexquadrilateralhavingdegreeoneandtwointeriorvertices(weightedFermat-
Torricelli points) having degree three.
Definition6. AweightedFermat-Torricellitreeofweightedminimumlengthwitha
Fermat-Torricelli tree topology of degree four is called a weighted Fermat-Torricelli
tree of degree four.
Definition 7. [5, Subsection 3.7,pp. 6],[17] A weighted Gauss tree of weighted
minimum length with a Gauss tree topology of degree three is called a generalized
Gauss tree of degree three or a full weighted minimal Steiner tree.
In 2014, we study an important generalization of the weighted Gauss (tree)
problem that we call a generalized Gauss problem for convex quadrilaterals in R2
by using a mechanical construction which extends the mechanical construction of
Gueron-Tessler in the sense of Po´lya and Varigon ([17, Problem 1, Theorem 4,
pp.1073-1075]).
4 ANASTASIOSN.ZACHOS
WestateageneralizedGaussproblemforaweightedconvexquadrilateralA A A A .
1 2 3 4
in R2, such that the weights Bi which correspondto Ai and B00′ ≡xG, satisfy the
inequalities
|B −B |<B <B +B ,
i j k i j
and
|B −B |<B <B +B
t m n t m
where x is the variable weight which corresponds to the given distance l ≡
G
kA0−A0′k, for i,j,k ∈{1,4,00′}, t,m,n∈{2,3,00′} and i6=j 6=k, t6=m6=n.
Problem 3. [17, Problem 1,p. 1073] Given l, B , B , B , B , find a generalized
1 2 3 4
Gauss tree of degree three with respect to A A A A which minimizes
1 2 3 4
B1kA1−A0k+B2kA2−A0k+B3kA3−A0′k+B4kA4−A0′k+xGl. (1.5)
For l =0, we obtain a weighted Fermat-Torricelli tree of degree four.
Definition 8. [17, Definition 8, p.1076] We call the variable x which depend on
G
l, a generalized Gauss variable.
Definition9. [17,Definition9,p.1080]Theresidualabsorbingrateofageneralized
Gauss tree of degree at most four with respect to a boundary convex quadrilateral is
4
B −x .
i G
i=1
X
Definition 10. [17, Definition 10, p.1080] An absorbing generalized Gauss tree of
degree three is a generalized Gauss tree of degree three with residual absorbing rate
4
B −x .
i G
i=1
X
In this paper, we prove that the weighted Fermat-Torricelli problem for convex
quadrilaterals cannot be solved analytically, by extending the geometrical solution
of E. Torricelli and E.Engelbrecht for convex quadrilaterals in R2 (Section 2, The-
orems 3,4)
In section 3, we derive a solution for the generalized Gauss problem in R2, in
the spirit of K. Menger which depends only on five given positive weighta and five
Euclideandistanceswhichdetermineaconvexquadrilateral(Section3,Theorem5).
Insection4,Wegiveanewapproachconcerningthedynamicplasticityofquadri-
lateralsbyderivinganewsystemofequationsdifferentfromthedynamicplasticity
equations which have been deduced in [13, Proposition 4.4,p. 417] and [16, Prob-
lem 2, Definition 12, Theorem 1,p.92,p.97-98] (Section 4, Proposition 2). Further-
more,we obtain a surprising connection of the dynamic plasticity of quadrilaterals
with a problem of Rene Descarted posed in 1638.
Insection5,weintroduceanabsorbingweightedFermat-Torricellitreeofdegree
four which is derived as a limiting tree structure from a generalized Gauss tree of
degree three of the same boundary convex quadrilateral in R2.
Then, by assuming that the four variable positive weights which correspond to
the fixed vertices of the boundary quadrilateral and satisfy the dynamic plasticity
SOLVING A PROBLEM OF ANGIOGENESIS 5
equations, we obtain a family of limiting tree structures of generalized Gauss trees
of degree three which concentrate to the same weighted Fermat-Torricelli tree of
degree four in a geometric sense (Universal absorbing Fermat-Torricelli tree).
Furthermore, we calculate the values of the universal rates of a Universal ab-
sorbing tree regarding a fixed boundary quadrilaterals (Section 5, Examples 2,3).
In section 6, we introduce a class of Euclidean minimal tree structures that
we call steady trees and evolutionary trees (Section 6, Definitions 15,16). Thus,
the minimum of the universal absorbing Fermat-Torricelli set (Universal Fermat-
Torricelliminimumvalue)leadstothecreationofageneralizedGausstreeofdegree
three for the same boundary convex quadrilateral which is derived by a weighted
Fermat-Torricellitree of degree four. A universal absorbing Fermat-Torricellimin-
imum value corresponds to the intersection point (4wFT point). This quantity is
of fundamental importance, because by attaining this value the absorbing Fermat-
Torricelli tree start to grow and will be able to produce a generalized Gauss tree
of degree three (Evolutionary tree). Each specific value from the universal absorb-
ing set gives an evolutionary process of a generalized Gauss tree of degree three
regarding a fixed boundary quadrilateralby spending a positive quantity from the
storage of the universal Fermat-Torricelli quantity which stimulates the evolution
at the 4wFT point (Section 6, Example 4, Angiogenesis of degree three).
2. Extending Torricelli-Engelbrecht’s solution for convex
quadrilaterals
The weighted Torricelli-Engelbrecht solution for a triangle △A A A in the
1 2 3
weighted floating case is given by the following proposition:
Lemma1. [6],[4]IfA isaninteriorweightedFermat-Torricellipointof△A A A ,
0 1 2 3
then
B2−B2−B2
∠A A A ≡α =arccos k i j . (2.1)
i 0 j i0j 2B B
i j !
Let A A A A be a convex quadrilateral in R2, O be the intersection point of
1 2 3 4
thetwodiagonalsandB be agivenweightwhichcorrespondstothe vertexA ,A
i i 0
be the weighted Fermat-Torricelli point in the weighted floating case (Theorem )
and U~ be the unit vector from A to A , for i,j =1,2,3,4.
ij i j
We mention the geometric plasticity principle of quadrilaterals in R2,
Lemma 2. [13],[16, Definition 13,Theorem 3, Proposition 8, Corollary 4 p. 103-
108] Suppose that the weighted floating case of the weighted Fermat-Torricelli point
A point with respect to A A A A is satisfied:
0 1 2 3 4
B U~ +B U~ +B U~ >B ,
i ki j kj m km k
(cid:13) (cid:13)
for each i,j,k,m = 1,2(cid:13),3,4 and i 6= j 6= k 6= m(cid:13). If A0 is connected with every
vertex A for k = 1,2,3(cid:13),4 and we select a point A(cid:13)′ with non-negative weight B
k k k
which lies on the ray A A and the quadrilateral A′A′A′A′ is constructed such
k 0 1 2 3 4
that:
BiU~k′i′ +BjU~k′j′ +BmU~k′m′ >Bk,
for each i′,j′,k′,m′ =(cid:13)1,2,3,4 and i′ 6= j′ 6= k′ 6= m(cid:13)′. Then the weighted Fermat-
(cid:13) (cid:13)
Torricelli point A′ is i(cid:13)dentical with A . (cid:13)
0 0
6 ANASTASIOSN.ZACHOS
Theorem 3. The weighted Torricelli-Engelbrecht solution for A A A A is given
1 2 3 4
by the following system of four equations w.r. to the variables α , α , α and
102 203 304
α :
401
csc2α csc2α csc2α (cosα −sinα )(cosα −sinα )
102 304 401 102 102 304 304
(cos(α −α )−cos(α +α +2α )−2sin(α +α ))=0,(2.2)
102 304 102 304 401 102 304
−B2−2B B cosα −B2+B2−2B B cosα −2B B cos(α +α )−B2 =0,(2.3)
1 1 2 102 2 3 1 4 401 2 4 102 401 4
B2+2B B cosα +B2−B2−B2
α =arccos 1 1 2 102 2 3 4 (2.4)
304 2B B
(cid:18) 3 4 (cid:19)
and
α =2π−α −α −α . (2.5)
203 102 304 401
Proof. Assume that we select B , B , B , B , such that A is an interior point
1 2 3 4 0
of △A OA . By applying the geometric plasticity principle of Lemma 2 we could
1 2
choose a transformation of A A A A to the square A′A′A′A′, where A′ = A
1 2 3 4 1 2 3 4 0 0
and A is an interior point of A′O′A′ where O′ is the intersection of A′A′ and
O 1 2 1 3
A′A′.
2 4
We consider the equations of the three circles which pass through A , A , A ,
1 2 0
A , A , A and A , A , A , respectively, which meet at A :
1 4 0 3 4 0 0
a 2 1 2 1
x− + y− acotα = a2csc2α (2.6)
102 102
2 2 4
(cid:16) (cid:17) (cid:18) (cid:19)
a 2 1 2 1
y− + x− acotα = a2csc2α (2.7)
401 401
2 2 4
(cid:16) (cid:17) (cid:18) (cid:19)
a 2 1 2 1
x− + y−(a− acotα ) = a2csc2α (2.8)
304 304
2 2 4
(cid:16) (cid:17) (cid:18) (cid:19)
By subtracting (2.7) from (2.6), (2.8) from(2.6) and solving w.r. to x,y we get:
a(−1+cotα )(−1+cotα )
x= 102 304 (2.9)
(−2+cotα +cotα )(−1+cotα )
102 304 401
and
−acotα +a
y =− 304 (2.10)
cotα +cotα −2
102 304
By substituting (2.9) and (2.10) in (2.6), we obtain (2.2).
Taking into account the weighted floating equilibrium condition, we get:
−B ~u =B ~u +B ~u +B ~u (2.11)
3 30 1 10 2 20 4 40
or
B ~u +B ~u =−B ~u −B ~u . (2.12)
1 10 2 20 3 30 4 40
By squaring both parts of (2.11) we derive (2.3) and by squaring both parts of
(2.12) we derive (2.4).
SOLVING A PROBLEM OF ANGIOGENESIS 7
Figure 1. A generalized Gauss Menger tree of degree three re-
garding a boundary convex quadrilateral
(cid:3)
Remark1. Adifferent approach was usedin[13,Solution2.2,Example2.4,p.413-
414], in order to derive a similar system of equations w.r. to α and α .
401 102
Example 1. By substituting B =3.5, B =2.5, B =2, B =1, a=10 in (2.2)
1 2 3 4
and (2.3) and solving this system of equations numerically by using for instance
Newton method and choosing as initial values αo = 2.7 rad, αo = 1.2 rad, we
102 401
obtain α = 2.30886 and α = 1.57801 rad. By substituting α = 2.30886
102 401 102
and α = 1.57801 rad in (2.4) we get α = 1.12492 rad. From (2.5), we get
401 304
α = 1.2714 rad. By substituting the angles α in (2.9) and (2.10), we derive
203 i0j
x=4.0700893 and y =2.146831.
Theorem 4. There does not exist an analytical solution for the 4wFT problem in
R2.
Proof. The system of the two equations (2.2) and (2.3) taking into account (2.4)
cannotbesolvedexplicitlyw.rtoα andα .Therefore,byconsidering(2.9)and
102 401
(2.10) we deduce that the location of the 4wFT point A cannot also be expressed
0
explicitly via the angles α , for i,j =1,2,3,4, for i6=j. (cid:3)
i0j
Thus,fromTheorem4thepositionofaweightedFermat-Torricellitreeofdegree
fourcannotbeexpressedanalyticallyandmaybefoundbyusingnumericalmethods
(see also in [13]).
3. An absorbing generalized Gauss-Menger tree in R2
Let A A A A be a boundary weighted convex quadrilateral of a generalized
1 2 3 4
GausstreeofdegreethreeinR2 andA0,A0′ arethetwoweightedFermat-Torricelli
(3wFT)pointsofdegreethreewhicharelocatedattheconvexhulloftheboundary
quadrilateral.
Wedenotebyl ≡kA0−A0′k,aij ≡kAi−Ajk,αijk ≡∠AiAjAk,a10 ≡a1,a40 ≡
a4, a20′ ≡ a2, a30′ ≡ a3 (Fig. 1) and by Bi ≡ P4iB=1i′Bi′, for i,j,k ∈ {0,0′,1,2,3,4}
and i6=j 6=k.
8 ANASTASIOSN.ZACHOS
Weproceedbygivingthefollowingtwolemmaswhichhavebeenprovedrecently
in [15, Theorem 1] and [14].
Lemma 3. [15, Theorem 1],[14, Theorem 2.1,p. 485]
A generalized Gauss tree of degree three (full weighted Steiner minimal tree)
of A A A A consists of two weighted Fermat-Torricelli points A , A′ which are
1 2 3 4 0 0
located at the interior convex domain with corresponding given weights B0, B0′ and
minimizes the objective function:
B a +B a +B a +B a + B0+B0′l→min, (3.1)
1 1 2 2 3 3 4 4
2
such that:
|B −B |<B <B +B , (3.2)
i j k i j
|B −B |<B <B +B (3.3)
t m n t m
where
B00′ ≡ B0+B0′,
2
for i,j,k∈{1,4,00′}, t,m,n∈{2,3,00′} and i6=j 6=k, t6=m6=n,
Suppose that B1, B2, B3, B4, B00′ satisfy the inequalities (3.2) and (3.3).
Lemma 4. [14, Theorem 2.2, p. 486] The location of A0 and A0′ is given by the
relations:
cotϕ= B0a12+B4a14cos(α214−α400′)+B3a23cos(α123−α30′0), (3.4)
B4a14sin(α214−α400′)−B3a23sin(α123−α30′0)
a = a14sin(α214−ϕ−α400′) (3.5)
1 sin(α100′ +α400′)
and
a = a23sin(α123+ϕ−α300′) (3.6)
2 sin(α20′0+α30′0)
where ϕ is the angle which is formed between the line defined by A and A and
1 2
the line which passes from A and it is parallel to the line defined by A and A′.
1 0 0
Definition 11. A generalized Gauss-Menger tree is a solution of a generalized
Gauss problem in R2 for a boundary quadrilateral A A A A which depend on the
1 2 3 4
Euclidean distances aij and the five given weights B1, B2, B3, B4 and B00′.
BylettingB00′ ≡xG,weobtainanabsorbinggeneralizedGauss-Mengertreefor
a boundary quadrilateral.
Theorem 5. An absorbing generalized Gauss-Menger tree w.r. to a fixed convex
quadrilateral A A A A depends only on the five given weights B , B , B , B ,
1 2 3 4 i 2 3 4
B00′ ≡xG and the five given lengths a12, a23, a34, a41 and a13.
Proof. Consider the Caley-Menger determinant which gives the volume of a tetra-
hedron A A A A in R3.
1 2 3 4
SOLVING A PROBLEM OF ANGIOGENESIS 9
0 a2 a2 a2 1
12 13 14
a2 0 a2 a2 1
 12 23 24 
288V2 =det a2 a2 0 a2 1 . (3.7)
13 23 34
 a2 a2 a2 0 1 
 14 24 34 
 1 1 1 1 0 
 
By letting V =0 in (3.7), weobtain a dependence of thesix distances a , a ,
12 13
a , a , a and a . For instance, by solving a fourth order degree equation w.r.
14 23 34 24
to a we derive that a =(a ,a ,a ,a ,a ).
13 13 12 14 23 34 24
By applying the cosine law in △A A A and △A A A we get:
1 2 4 1 2 3
a2 +a2 −a2
α =arccos 12 14 24 , (3.8)
214 2a a
(cid:18) 12 14 (cid:19)
and
a2 +a2 −a2
α =arccos 12 23 13 . (3.9)
123 2a a
(cid:18) 12 23 (cid:19)
By Lemma 1 and Lemma 3 and considering that A is the 3wFT point of
0
△A1A4A0′ and A0′ is the 3wFT point of △A2A3A0 we get:
B2−B2−x2
α100′ =arccos 4 2B 1x G , (3.10)
(cid:18) 1 G (cid:19)
B2−B2−x2
α0′04 =arccos 1 2B 4x G , (3.11)
(cid:18) 4 G (cid:19)
x2 −B2−B2
α =arccos G 1 4 , (3.12)
104 2B B
(cid:18) 1 4 (cid:19)
B2−B2−x2
α00′3 =arccos 2 2B 3x G , (3.13)
(cid:18) 3 G (cid:19)
B2−x2 −B2
α00′2 =arccos 3 2x GB 2 , (3.14)
(cid:18) G 2 (cid:19)
and
x2 −B2−B2
α20′3 =arccos G 2B 2B 3 . (3.15)
(cid:18) 2 3 (cid:19)
Therefore, by replacing (3.10), (3.11), (3.14), (3.13), (3.8), (3.9) in (3.4), (3.5),
(3.6) and taking into account the dependence of the six distances a , for i,j =
ij
1,2,3,4, we derive that ϕ, a and a depend only on B , B , B , B , x and a ,
1 2 1 2 3 4 G 12
a , a , a , a .
13 23 34 24
(cid:3)
4. The dynamic plasticity of convex quadrilaterals
In this section, we deal with the solution of the inverse 4wFT problem in R2
which has been introduced in [13] and developed in [16], in order to obtain a new
systemofequationsofthedynamicplasticityofweightedquadrilateralsw.r. tothe
fourvariableweights(B ) fori=1,2,3,4,whichcoveralsothe case(B ) =
i 1234 1 1234
(B ) and (B ) =(B ) .
3 1234 2 1234 4 1234
First, we start by mentioning the solution of S. Gueron and R. Tessler ([6, Sec-
tion4,p.449])oftheinverse3wFTproblemforthreenon-collinearpointsinR2.By
10 ANASTASIOSN.ZACHOS
letting B =0 in the inverse 4wFT problem for convex quadrilaterals (Problem 2)
i
we derive the inverse 3wFT problem for s triangle.
Consider the inverse 3wFT problem for △A A A in R2.
i j k
Lemma 5. [6, Section 4,p. 449] The unique solution of the inverse 3wFT problem
for △A A A is given by
i j k
B sinα
i jik
( ) = . (4.1)
ijk
B sinα
j ijk
Definition 12. [16] We call dynamic plasticity of a weighted Fermat-Torricelli
tree of degree four the set of solutions of the four variable weights with respect to
the inverse 4wFT problem in R2 for a given constant value c which correspond to
a family of weighted Fermat-Torricelli tree of degree four that preserve the same
Euclidean tree structure (the corresponding 4wFT point remains the same for a
fixed boundary convex quadrilateral), such that the three variable weights depend on
a fourth variable weight and the value of c.
BytakingintoaccountLemma5forthetriangles△A A A ,△A A A ,△A A A
1 2 3 1 3 4 1 2 4
and the weighted floating equilibrium condition (1.3) taken from Theorem 1 ([13],
[16])
Proposition 1. [13, Proposition 4.4,p. 417],[16, Problem 2, Definition 12, The-
orem 1,p.92,p.97-98] Suppose that A does not belong to the intersection of the
0
linear segments A A and A A . The dynamic plasticity of the variable weighted
1 3 2 4
Fermat-Torricelli tree of degree four in R2 is given by the following three equations:
B B B B
( 2) =( 2) [1−( 4) ( 1) ], (4.2)
B 1234 B 123 B 1234 B 134
1 1 1 4
B B B B
( 3) =( 3) [1−( 4) ( 1) ], (4.3)
B 1234 B 123 B 1234 B 124
1 1 1 4
and
(B ) +(B ) +(B ) +(B ) =c=const. (4.4)
1 1234 2 1234 3 1234 4 1234
Itisworthmentioningthateachquadofvalues{(B ) ,(B ) ,(B ) ,(B ) },
1 1234 2 1234 3 1234 4 1234
whichsatisfysimultaneously(4.2), (4.3), (4.4)createaunique concentrationofdif-
ferentfamiliesoftetrafocalellipses(PolyellipseorEgglipse)tothesame4wFTpoint
A of A A A A , A family of tetrafocal ellipses may be constructed by selecting a
0 1 2 3 4
decreasing sequence of real numbers c ((B ) ,(B ) ,(B ) ,(B ) ))
n 1 1234 2 1234 3 1234 4 1234
4
c ((B ) ,(B ) ,(B ) ,(B ) );X)= (B ) kA −Xk
n 1 1234 2 1234 3 1234 4 1234 i 1234 i
i=1
X
which converge to the constant number
c((B ) ,(B ) ,(B ) ,(B ) )≡f(A ,(B ) ,(B ) ,(B ) ,(B ) ).
1 1234 2 1234 3 1234 4 1234 0 1 1234 2 1234 3 1234 4 1234
or
4
c((B ) ,(B ) ,(B ) ,(B ) )= (B ) kA −A k
1 1234 2 1234 3 1234 4 1234 i 1234 i 0
i=1
X